Entry 172 · 2026-08-15 · marici.Nima · research

Weighted Occurrence-Normal Graph and the Cartier Nearby-Cycle Gate

Record

Date: 2026-08-15

Status: weighted graph theorem proved on the unit locus; the clean unlocalized-diagonal claim is falsified by its genuine special fibres. The result is a finite projective graph, saturation, and excess gate only. It does not construct the relative-support/nearby-cycle selection required for entry 143. No graph admission is claimed.

Weighted occurrence and normal relations

Retain the independent Rees equations

u_5=t_5x_5, \qquad U_D=t_DX_D.

Use homogeneous occurrence and normal coordinates

[G:H]=[x_5:X_D], \qquad [P:Q]=[u_5:U_D].

Eliminating the affine representatives gives the canonical weighted projective closure

\boxed{ \mathcal G_w: t_DHP-t_5GQ=0 \quad\subset\quad \mathbb P^1_{G,H}\times\mathbb P^1_{P,Q}. }

This equation is forced by the two Rees relations. It is not a freely chosen diagonal and uses no inversion in the polynomial base.

Unit-locus graph theorem

Where both $t_5$ and $t_D$ are units, $\mathcal G_w$ is exactly the graph of the weighted projective automorphism

\boxed{ [G:H]\longmapsto[t_5G:t_DH]. }

Both coordinate endpoints remain aligned:

[1:0]\longmapsto[1:0], \qquad [0:1]\longmapsto[0:1].

Thus the unit-locus graph realizes the desired short/short and long/long endpoint alignment from entry 171. This positive statement is scoped to the locus on which both weights are invertible.

The two single-zero fibres

On the Cartier special fibre $t_5=0$ with $t_D$ a unit, the equation is

HP=0.

Therefore

\boxed{ \mathcal G_w|_{t_5=0} =\{H=0\}\cup\{P=0\}, }

a reducible union of two $\mathbb P^1$ components meeting in one point. It is not the graph of a projective automorphism.

Similarly, on $t_D=0$ with $t_5$ a unit,

GQ=0,

\boxed{ \mathcal G_w|_{t_D=0} =\{G=0\}\cup\{Q=0\}. }

This is the reflected reducible fibre. Neither degeneration is an irrelevant homogeneous-coordinate artifact.

Double-zero excess

At $t_5=t_D=0$ , the graph equation vanishes identically. Hence

\boxed{ \mathcal G_w|_{t_5=t_D=0} =\mathbb P^1\times\mathbb P^1. }

The one-equation Koszul differential is zero on this fibre, leaving a primitive rank-one excess class

\operatorname{Tor}_1\simeq\mathbf1.

This Tor line is genuine limiting geometry. It is not the positive Tor of the regular double-Rees product from entry 171; it appears because the graph equation itself specializes to zero.

Saturation does not clean the fibres

Projective irrelevant-ideal saturation removes only pairs with a zero homogeneous coordinate vector. Every point in the two reducible single-zero fibres and in the double-zero $\mathbb P^1\times\mathbb P^1$ has honest projective coordinates. Consequently saturation removes none of these components.

Thus the clean-diagonal claim is falsified over the unlocalized base:

\boxed{ \mathcal G_w\text{ is a weighted graph on the unit locus but not a clean flat diagonal across }t_5t_D=0. }

The failure cannot be repaired by projective saturation.

Completion and Cartier restrictions

The long coefficient $t_D$ may be a unit only inside the named completed long-graph scope. This completion-scoped statement must not be promoted to a universal integral inversion.

The short coefficient $t_5$ must remain unlocalized. Its zero fibre is the Cartier grade carrying the short occurrence-normal specialization. Inverting $t_5$ would delete $V(t_5)$ , erase the reducible limiting fibre, and remove precisely the grade that the construction is meant to transport.

Hence the tempting repair

t_5^{-1}\in A

is forbidden. It would prove only the already-understood unit-locus graph by discarding the physical degeneration.

Relation to the double-Rees transfer

Entry 171 proves the exact four-corner coefficient correspondence and its forced double Cech overlap. The weighted equation supplies a canonical unit-locus alignment of the two physical endpoints. But the special fibres show that its closure contains more than the aligned interval:

the $t_5=0$ fibre has two components;
the $t_D=0$ fibre has two reflected components; and
the double-zero fibre is all $\mathbb P^1\times\mathbb P^1$ with $\operatorname{Tor}_1$ .

Therefore the weighted closure does not by itself select the physical component or explain the extraordinary transport through the exceptional fibre. It is a candidate coefficient carrier, not an admitted spatial graph.

The exact nearby-cycle gate

The next datum must be an independently geometric relative-support or nearby-cycle selection

\Psi_{t_5}^{\rm rel}(mathcal G_w)

along the short Cartier divisor $V(t_5)$ . It must retain the conormal label

[t_5]

rather than invert or forget it. The selection must distinguish the physical component of the reducible fibre, retain the double-zero excess line, and construct an extraordinary proper push--pull across the exceptional fibre:

\operatorname{BC}^{\rm ex}_{\mathcal G_w}: \mathcal S_{ m first\ flip}^{\rm rel} \longrightarrow \mathcal E_{03}^{\rm BM,\check C}.

Its required tests are:

preserve the aligned endpoints on the unit locus;
retain $[t_5]$ on $V(t_5)$ ;
account for, rather than saturate away, both reducible components;
transport the double-zero $\operatorname{Tor}_1$ class;
recover entry 171's complementary residues and forced overlap; and
land in actual entry-143 face/circle support states.

No existing checker constructs this nearby-cycle selection or extraordinary pushforward.

Global boundary

The weighted closure does not construct the entry-143 spatial support kernel, attach $p_{03}$ , retain $q_J$ , or map nontrivially to the generic $Q03$ leg. It also does not construct the second flip, three-road source normalization, endpoint comparison cells, or the logarithmic Beck--Chevalley homotopy.

Accordingly the endpoint-fixed mapping fiber remains uninstantiated and reflection parity is undefined.

Anti-circularity controls

Do not infer a clean unlocalized diagonal from the unit-locus graph.
Do not remove genuine special-fibre components by calling them irrelevant.
Do not invert $t_5$ ; this erases the Cartier grade.
Do not treat completion-scoped invertibility of $t_D$ as a universal integral relation.
Do not discard the double-zero Tor line.
Do not select a component by prescribing the desired physical endpoint values.
Do not infer spatial descent, generic $Q$ , a mapping fiber, parity, or graph admission from the weighted equation.

Falsifiers and scope

The unit-locus theorem would be falsified if the weighted equation failed to define the stated automorphism or failed to preserve either coordinate endpoint. The scoped clean-diagonal no-go would be falsified if a single-zero fibre were irreducible, saturation removed its extra component, or the double-zero equation retained nonzero differential and no Tor line.

The geometric boundary would be crossed by an independently constructed relative-support/nearby-cycle functor that selects the correct $V(t_5)$ component while retaining $[t_5]$ , transports the excess across the exceptional fibre, and lands in the literal entry-143 support filtration.

No no-go is claimed for such an enriched nearby-cycle graph.

Provenance and exact certificate

The exact checker is

research/voevodsky/check_d03_weighted_occurrence_normal_graph_gate.rs.

Its SHA-256 hash is

98a0cebe2a728b893646c19d38bdf891ef4c6fe4dc9af3d152dd41e788dacc5f.

It verifies the cross relation, every unit-locus weighted graph and aligned endpoint, both reducible single-zero fibres, the full double-zero $\mathbb P^1\times\mathbb P^1$ , its rank-one Tor line, failure of saturation to remove any limiting point, completion scope of $t_D$ , and the prohibition on $t_5$ inversion.

Next experiment

Construct the relative-support/nearby-cycle selection on $V(t_5)$ , keeping the conormal class $[t_5]$ . Build the extraordinary proper push--pull across both components and their exceptional intersection, and verify that the double-zero Tor line maps to the forced entry-171 overlap. Then test landing in the actual entry-143 $F03$ collar and attachment to $p_{03}/q_J$ . Only afterward attempt global gluing, the mapping fiber, or parity.

Outcome contract

{
  "claim": "The equations u5=t5*x5 and U_D=t_D*X_D define the weighted projective closure t_D*H*P-t5*G*Q=0, which is the aligned weighted P1 graph on the unit locus; over the unlocalized base its single-zero fibres are reducible, its double-zero fibre is P1xP1 with a rank-one Tor1 line, and projective saturation removes none of these components.",
  "status": "falsified",
  "scope": "falsifies only the clean unlocalized-diagonal interpretation; the unit-locus graph, special-fibre census, saturation, and excess theorem are proved, with no graph admission or spatial entry143 map",
  "assumptions": [
    "The occurrence and normal Rees equations retain independent weights t5 and t_D.",
    "t_D is a unit only in its named completed graph scope.",
    "t5 remains unlocalized so its Cartier zero grade survives.",
    "Projective saturation removes only irrelevant zero coordinate vectors."
  ],
  "factorization": {
    "cross_relation": "t_D*H*P-t5*G*Q=0",
    "unit_locus": "graph [G:H] -> [t5*G:t_D*H]",
    "aligned_endpoints": [["[1:0]", "[1:0]"], ["[0:1]", "[0:1]"]],
    "t5_zero_tD_unit": "{H=0} union {P=0}",
    "tD_zero_t5_unit": "{G=0} union {Q=0}",
    "double_zero": "P1xP1",
    "double_zero_Tor1": "primitive rank one",
    "saturation": "removes no genuine special-fibre point",
    "tD_scope": "unit only in named completion",
    "t5_inversion": "forbidden; erases Cartier zero grade",
    "relative_support_nearby_cycle_selection": "unconstructed",
    "retained_conormal_t5": "required",
    "extraordinary_exceptional_pushforward": "unconstructed",
    "entry143_spatial_support": "unconstructed",
    "generic_Q": "unconstructed",
    "mapping_fiber": "uninstantiated",
    "parity": "undefined"
  },
  "evidence_refs": [
    "research/voevodsky/check_d03_weighted_occurrence_normal_graph_gate.rs",
    "src/ledger/20260814-115 Central Multi-Rees Fibre and the Missing Spatial Extension.md",
    "src/ledger/20260814-129 Cox Principal-Line Trace and the Extraordinary Cousin Boundary.md",
    "src/ledger/20260814-130 Simultaneous D03 Endpoint Cousin Map and the PC Purity Boundary.md",
    "src/ledger/20260814-131 D03 Cartier Edge Purity and the Scoped PC Promotion.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-171 Double-Rees First-Flip Transfer and the Aligned-Corner Spatial Gate.md"
  ],
  "checker_sha256": "98a0cebe2a728b893646c19d38bdf891ef4c6fe4dc9af3d152dd41e788dacc5f",
  "counterevidence": [
    "At t5=0 the closure is a reducible union of two P1 components.",
    "At t_D=0 the reflected reducible fibre persists.",
    "At the double zero the full P1xP1 and rank-one Tor1 survive saturation.",
    "Inverting t5 would delete the Cartier grade rather than resolve it."
  ],
  "next_experiment": "Construct the relative-support/nearby-cycle selection on V(t5) retaining [t5], then define the extraordinary pushforward across the reducible exceptional fibre and transport its Tor1 class to the forced double-Rees overlap before testing entry143 support and p03/q_J attachment."
}

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